\(\int \frac {x^6 (e+f x^2)}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx\) [211]

Optimal result

Integrand size = 34, antiderivative size = 468 \[ \int \frac {x^6 \left (e+f x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=-\frac {(25 a b c d f-4 (b c-a d) (7 b d e-6 b c f+6 a d f)) x \sqrt {a-b x^2} \sqrt {c+d x^2}}{105 b^3 d^3}-\frac {(7 b d e-6 b c f+6 a d f) x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{35 b^2 d^2}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}+\frac {\sqrt {a} \left (48 a^3 d^3 f-a b^2 c d (49 d e-40 c f)+8 b^3 c^2 (7 d e-6 c f)+8 a^2 b d^2 (7 d e-5 c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{105 b^{7/2} d^4 \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}-\frac {\sqrt {a} c \left (24 a^3 d^3 f+a^2 b d^2 (28 d e-17 c f)-a b^2 c d (21 d e-16 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{105 b^{7/2} d^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:

-1/105*(25*a*b*c*d*f-4*(-a*d+b*c)*(6*a*d*f-6*b*c*f+7*b*d*e))*x*(-b*x^2+a)^ 
(1/2)*(d*x^2+c)^(1/2)/b^3/d^3-1/35*(6*a*d*f-6*b*c*f+7*b*d*e)*x^3*(-b*x^2+a 
)^(1/2)*(d*x^2+c)^(1/2)/b^2/d^2-1/7*f*x^5*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2) 
/b/d+1/105*a^(1/2)*(48*a^3*d^3*f-a*b^2*c*d*(-40*c*f+49*d*e)+8*b^3*c^2*(-6* 
c*f+7*d*e)+8*a^2*b*d^2*(-5*c*f+7*d*e))*(1-b*x^2/a)^(1/2)*(d*x^2+c)^(1/2)*E 
llipticE(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/b^(7/2)/d^4/(-b*x^2+a)^(1/2)/ 
(1+d*x^2/c)^(1/2)-1/105*a^(1/2)*c*(24*a^3*d^3*f+a^2*b*d^2*(-17*c*f+28*d*e) 
-a*b^2*c*d*(-16*c*f+21*d*e)+8*b^3*c^2*(-6*c*f+7*d*e))*(1-b*x^2/a)^(1/2)*(1 
+d*x^2/c)^(1/2)*EllipticF(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/b^(7/2)/d^4/ 
(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.94 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.84 \[ \int \frac {x^6 \left (e+f x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {-\frac {b}{a}} d x \left (-a+b x^2\right ) \left (c+d x^2\right ) \left (24 a^2 d^2 f+a b d \left (28 d e-23 c f+18 d f x^2\right )+b^2 \left (24 c^2 f+3 d^2 x^2 \left (7 e+5 f x^2\right )-2 c d \left (14 e+9 f x^2\right )\right )\right )-i c \left (48 a^3 d^3 f+8 a^2 b d^2 (7 d e-5 c f)-8 b^3 c^2 (-7 d e+6 c f)+a b^2 c d (-49 d e+40 c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i c \left (24 a^3 d^3 f+a^2 b d^2 (28 d e-17 c f)-8 b^3 c^2 (-7 d e+6 c f)+a b^2 c d (-21 d e+16 c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{105 b^3 \sqrt {-\frac {b}{a}} d^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(x^6*(e + f*x^2))/(Sqrt[a - b*x^2]*Sqrt[c + d*x^2]),x]
 

Output:

(Sqrt[-(b/a)]*d*x*(-a + b*x^2)*(c + d*x^2)*(24*a^2*d^2*f + a*b*d*(28*d*e - 
 23*c*f + 18*d*f*x^2) + b^2*(24*c^2*f + 3*d^2*x^2*(7*e + 5*f*x^2) - 2*c*d* 
(14*e + 9*f*x^2))) - I*c*(48*a^3*d^3*f + 8*a^2*b*d^2*(7*d*e - 5*c*f) - 8*b 
^3*c^2*(-7*d*e + 6*c*f) + a*b^2*c*d*(-49*d*e + 40*c*f))*Sqrt[1 - (b*x^2)/a 
]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] 
 + I*c*(24*a^3*d^3*f + a^2*b*d^2*(28*d*e - 17*c*f) - 8*b^3*c^2*(-7*d*e + 6 
*c*f) + a*b^2*c*d*(-21*d*e + 16*c*f))*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2) 
/c]*EllipticF[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(105*b^3*Sqrt[-( 
b/a)]*d^4*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {444, 444, 444, 399, 323, 323, 321, 331, 330, 327}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^6 \left (e+f x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\int \frac {x^4 \left ((7 b d e-6 b c f+6 a d f) x^2+5 a c f\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\frac {\int \frac {x^2 \left (\left (-4 c (7 d e-6 c f) b^2+a d (28 d e-23 c f) b+24 a^2 d^2 f\right ) x^2+3 a c (7 b d e-6 b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (8 c^2 (7 d e-6 c f) b^3-a c d (49 d e-40 c f) b^2+8 a^2 d^2 (7 d e-5 c f) b+48 a^3 d^3 f\right ) x^2+a c \left (-4 c (7 d e-6 c f) b^2+a d (28 d e-23 c f) b+24 a^2 d^2 f\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\frac {\frac {\frac {\left (48 a^3 d^3 f+8 a^2 b d^2 (7 d e-5 c f)-a b^2 c d (49 d e-40 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \left (24 a^3 d^3 f+a^2 b d^2 (28 d e-17 c f)-a b^2 c d (21 d e-16 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{d}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {\frac {\frac {\frac {\left (48 a^3 d^3 f+8 a^2 b d^2 (7 d e-5 c f)-a b^2 c d (49 d e-40 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \sqrt {\frac {d x^2}{c}+1} \left (24 a^3 d^3 f+a^2 b d^2 (28 d e-17 c f)-a b^2 c d (21 d e-16 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {1}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {c+d x^2}}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {\frac {\frac {\frac {\left (48 a^3 d^3 f+8 a^2 b d^2 (7 d e-5 c f)-a b^2 c d (49 d e-40 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \left (24 a^3 d^3 f+a^2 b d^2 (28 d e-17 c f)-a b^2 c d (21 d e-16 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {\frac {\frac {\left (48 a^3 d^3 f+8 a^2 b d^2 (7 d e-5 c f)-a b^2 c d (49 d e-40 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \left (24 a^3 d^3 f+a^2 b d^2 (28 d e-17 c f)-a b^2 c d (21 d e-16 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\)

\(\Big \downarrow \) 331

\(\displaystyle \frac {\frac {\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \left (48 a^3 d^3 f+8 a^2 b d^2 (7 d e-5 c f)-a b^2 c d (49 d e-40 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \left (24 a^3 d^3 f+a^2 b d^2 (28 d e-17 c f)-a b^2 c d (21 d e-16 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\)

\(\Big \downarrow \) 330

\(\displaystyle \frac {\frac {\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (48 a^3 d^3 f+8 a^2 b d^2 (7 d e-5 c f)-a b^2 c d (49 d e-40 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {\sqrt {\frac {d x^2}{c}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \left (24 a^3 d^3 f+a^2 b d^2 (28 d e-17 c f)-a b^2 c d (21 d e-16 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (48 a^3 d^3 f+8 a^2 b d^2 (7 d e-5 c f)-a b^2 c d (49 d e-40 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \left (24 a^3 d^3 f+a^2 b d^2 (28 d e-17 c f)-a b^2 c d (21 d e-16 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\)

Input:

Int[(x^6*(e + f*x^2))/(Sqrt[a - b*x^2]*Sqrt[c + d*x^2]),x]
 

Output:

-1/7*(f*x^5*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(b*d) + (-1/5*((7*b*d*e - 6*b 
*c*f + 6*a*d*f)*x^3*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(b*d) + (-1/3*(((24*a 
^2*d*f)/b + a*(28*d*e - 23*c*f) - (4*b*c*(7*d*e - 6*c*f))/d)*x*Sqrt[a - b* 
x^2]*Sqrt[c + d*x^2]) + ((Sqrt[a]*(48*a^3*d^3*f - a*b^2*c*d*(49*d*e - 40*c 
*f) + 8*b^3*c^2*(7*d*e - 6*c*f) + 8*a^2*b*d^2*(7*d*e - 5*c*f))*Sqrt[1 - (b 
*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b 
*c))])/(Sqrt[b]*d*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) - (Sqrt[a]*c*(24*a^ 
3*d^3*f + a^2*b*d^2*(28*d*e - 17*c*f) - a*b^2*c*d*(21*d*e - 16*c*f) + 8*b^ 
3*c^2*(7*d*e - 6*c*f))*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[A 
rcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*Sqrt[a - b*x^2]*Sq 
rt[c + d*x^2]))/(3*b*d))/(5*b*d))/(7*b*d)
 

Defintions of rubi rules used

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 323

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( 
d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 330

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]   Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ[a, 
0]
 

rule 331

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 444

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ 
(p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ 
(b*d*(m + 2*(p + q + 1) + 1))   Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) 
^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( 
m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, 
q}, x] && GtQ[m, 1]
 

Maple [A] (verified)

Time = 10.77 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.26

method	result	size
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {f \,x^{5} \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{7 b d}-\frac {\left (e +\frac {f \left (6 a d -6 b c \right )}{7 b d}\right ) x^{3} \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{5 b d}-\frac {\left (\frac {5 a c f}{7 b d}+\frac {\left (e +\frac {f \left (6 a d -6 b c \right )}{7 b d}\right ) \left (4 a d -4 b c \right )}{5 b d}\right ) x \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{3 b d}+\frac {\left (\frac {5 a c f}{7 b d}+\frac {\left (e +\frac {f \left (6 a d -6 b c \right )}{7 b d}\right ) \left (4 a d -4 b c \right )}{5 b d}\right ) a c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{3 b d \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {\left (\frac {3 \left (e +\frac {f \left (6 a d -6 b c \right )}{7 b d}\right ) a c}{5 b d}+\frac {\left (\frac {5 a c f}{7 b d}+\frac {\left (e +\frac {f \left (6 a d -6 b c \right )}{7 b d}\right ) \left (4 a d -4 b c \right )}{5 b d}\right ) \left (2 a d -2 b c \right )}{3 b d}\right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(590\)
risch	\(-\frac {x \left (15 f \,x^{4} b^{2} d^{2}+18 a b \,d^{2} f \,x^{2}-18 b^{2} c f \,x^{2} d +21 b^{2} d^{2} e \,x^{2}+24 f \,d^{2} a^{2}-23 f d c b a +28 a b \,d^{2} e +24 f \,c^{2} b^{2}-28 d \,b^{2} c e \right ) \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{105 b^{3} d^{3}}+\frac {\left (-\frac {\left (48 f \,d^{3} a^{3}-40 a^{2} b c \,d^{2} f +56 a^{2} b \,d^{3} e +40 a \,b^{2} c^{2} d f -49 a \,b^{2} c \,d^{2} e -48 b^{3} c^{3} f +56 b^{3} c^{2} d e \right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}+\frac {24 a \,b^{2} c^{3} f \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {24 a^{3} c \,d^{2} f \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {28 a \,b^{2} c^{2} d e \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {28 a^{2} b c \,d^{2} e \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {23 a^{2} b \,c^{2} d f \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{105 b^{3} d^{3} \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(856\)
default	\(\text {Expression too large to display}\)	\(1327\)

Input:

int(x^6*(f*x^2+e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE 
)
 

Output:

((-b*x^2+a)*(d*x^2+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-1/7/b/d*f* 
x^5*(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)-1/5*(e+1/7/b/d*f*(6*a*d-6*b*c))/b 
/d*x^3*(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)-1/3*(5/7*a/b*c/d*f+1/5*(e+1/7/ 
b/d*f*(6*a*d-6*b*c))/b/d*(4*a*d-4*b*c))/b/d*x*(-b*d*x^4+a*d*x^2-b*c*x^2+a* 
c)^(1/2)+1/3*(5/7*a/b*c/d*f+1/5*(e+1/7/b/d*f*(6*a*d-6*b*c))/b/d*(4*a*d-4*b 
*c))/b/d*a*c/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(-b*d*x^4+a*d 
*x^2-b*c*x^2+a*c)^(1/2)*EllipticF(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))- 
(3/5*(e+1/7/b/d*f*(6*a*d-6*b*c))/b/d*a*c+1/3*(5/7*a/b*c/d*f+1/5*(e+1/7/b/d 
*f*(6*a*d-6*b*c))/b/d*(4*a*d-4*b*c))/b/d*(2*a*d-2*b*c))*c/(b/a)^(1/2)*(1-b 
*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)/d*(El 
lipticF(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))-EllipticE(x*(b/a)^(1/2),(- 
1-(a*d-b*c)/c/b)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.06 \[ \int \frac {x^6 \left (e+f x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {-b d} {\left (7 \, {\left (8 \, a b^{3} c^{2} d - 7 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} e - 8 \, {\left (6 \, a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 5 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3}\right )} f\right )} x \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) - \sqrt {-b d} {\left (7 \, {\left (8 \, a^{3} b d^{3} + 4 \, {\left (2 \, a b^{3} - b^{4}\right )} c^{2} d - {\left (7 \, a^{2} b^{2} - 4 \, a b^{3}\right )} c d^{2}\right )} e + {\left (48 \, a^{4} d^{3} - 24 \, {\left (2 \, a b^{3} - b^{4}\right )} c^{3} + {\left (40 \, a^{2} b^{2} - 23 \, a b^{3}\right )} c^{2} d - 8 \, {\left (5 \, a^{3} b - 3 \, a^{2} b^{2}\right )} c d^{2}\right )} f\right )} x \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) + {\left (15 \, b^{4} d^{3} f x^{6} + 3 \, {\left (7 \, b^{4} d^{3} e - 6 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} f\right )} x^{4} - {\left (28 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} e - {\left (24 \, b^{4} c^{2} d - 23 \, a b^{3} c d^{2} + 24 \, a^{2} b^{2} d^{3}\right )} f\right )} x^{2} + 7 \, {\left (8 \, b^{4} c^{2} d - 7 \, a b^{3} c d^{2} + 8 \, a^{2} b^{2} d^{3}\right )} e - 8 \, {\left (6 \, b^{4} c^{3} - 5 \, a b^{3} c^{2} d + 5 \, a^{2} b^{2} c d^{2} - 6 \, a^{3} b d^{3}\right )} f\right )} \sqrt {-b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, b^{5} d^{4} x} \] Input:

integrate(x^6*(f*x^2+e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="fri 
cas")
 

Output:

-1/105*(sqrt(-b*d)*(7*(8*a*b^3*c^2*d - 7*a^2*b^2*c*d^2 + 8*a^3*b*d^3)*e - 
8*(6*a*b^3*c^3 - 5*a^2*b^2*c^2*d + 5*a^3*b*c*d^2 - 6*a^4*d^3)*f)*x*sqrt(a/ 
b)*elliptic_e(arcsin(sqrt(a/b)/x), -b*c/(a*d)) - sqrt(-b*d)*(7*(8*a^3*b*d^ 
3 + 4*(2*a*b^3 - b^4)*c^2*d - (7*a^2*b^2 - 4*a*b^3)*c*d^2)*e + (48*a^4*d^3 
 - 24*(2*a*b^3 - b^4)*c^3 + (40*a^2*b^2 - 23*a*b^3)*c^2*d - 8*(5*a^3*b - 3 
*a^2*b^2)*c*d^2)*f)*x*sqrt(a/b)*elliptic_f(arcsin(sqrt(a/b)/x), -b*c/(a*d) 
) + (15*b^4*d^3*f*x^6 + 3*(7*b^4*d^3*e - 6*(b^4*c*d^2 - a*b^3*d^3)*f)*x^4 
- (28*(b^4*c*d^2 - a*b^3*d^3)*e - (24*b^4*c^2*d - 23*a*b^3*c*d^2 + 24*a^2* 
b^2*d^3)*f)*x^2 + 7*(8*b^4*c^2*d - 7*a*b^3*c*d^2 + 8*a^2*b^2*d^3)*e - 8*(6 
*b^4*c^3 - 5*a*b^3*c^2*d + 5*a^2*b^2*c*d^2 - 6*a^3*b*d^3)*f)*sqrt(-b*x^2 + 
 a)*sqrt(d*x^2 + c))/(b^5*d^4*x)
 

Sympy [F]

\[ \int \frac {x^6 \left (e+f x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^{6} \left (e + f x^{2}\right )}{\sqrt {a - b x^{2}} \sqrt {c + d x^{2}}}\, dx \] Input:

integrate(x**6*(f*x**2+e)/(-b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)
 

Output:

Integral(x**6*(e + f*x**2)/(sqrt(a - b*x**2)*sqrt(c + d*x**2)), x)
 

Maxima [F]

\[ \int \frac {x^6 \left (e+f x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )} x^{6}}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:

integrate(x^6*(f*x^2+e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="max 
ima")
 

Output:

integrate((f*x^2 + e)*x^6/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)), x)
 

Giac [F]

\[ \int \frac {x^6 \left (e+f x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )} x^{6}}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:

integrate(x^6*(f*x^2+e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="gia 
c")
 

Output:

integrate((f*x^2 + e)*x^6/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6 \left (e+f x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^6\,\left (f\,x^2+e\right )}{\sqrt {a-b\,x^2}\,\sqrt {d\,x^2+c}} \,d x \] Input:

int((x^6*(e + f*x^2))/((a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)),x)
 

Output:

int((x^6*(e + f*x^2))/((a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^6 \left (e+f x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:

int(x^6*(f*x^2+e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
 

Output:

( - 24*sqrt(c + d*x**2)*sqrt(a - b*x**2)*a**2*d**2*f*x + 23*sqrt(c + d*x** 
2)*sqrt(a - b*x**2)*a*b*c*d*f*x - 28*sqrt(c + d*x**2)*sqrt(a - b*x**2)*a*b 
*d**2*e*x - 18*sqrt(c + d*x**2)*sqrt(a - b*x**2)*a*b*d**2*f*x**3 - 24*sqrt 
(c + d*x**2)*sqrt(a - b*x**2)*b**2*c**2*f*x + 28*sqrt(c + d*x**2)*sqrt(a - 
 b*x**2)*b**2*c*d*e*x + 18*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b**2*c*d*f*x* 
*3 - 21*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b**2*d**2*e*x**3 - 15*sqrt(c + d 
*x**2)*sqrt(a - b*x**2)*b**2*d**2*f*x**5 + 48*int((sqrt(c + d*x**2)*sqrt(a 
 - b*x**2)*x**2)/(a*c + a*d*x**2 - b*c*x**2 - b*d*x**4),x)*a**3*d**3*f - 4 
0*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x**2 - b*c*x**2 
- b*d*x**4),x)*a**2*b*c*d**2*f + 56*int((sqrt(c + d*x**2)*sqrt(a - b*x**2) 
*x**2)/(a*c + a*d*x**2 - b*c*x**2 - b*d*x**4),x)*a**2*b*d**3*e + 40*int((s 
qrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x**2 - b*c*x**2 - b*d*x* 
*4),x)*a*b**2*c**2*d*f - 49*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/( 
a*c + a*d*x**2 - b*c*x**2 - b*d*x**4),x)*a*b**2*c*d**2*e - 48*int((sqrt(c 
+ d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x**2 - b*c*x**2 - b*d*x**4),x) 
*b**3*c**3*f + 56*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c + a*d* 
x**2 - b*c*x**2 - b*d*x**4),x)*b**3*c**2*d*e + 24*int((sqrt(c + d*x**2)*sq 
rt(a - b*x**2))/(a*c + a*d*x**2 - b*c*x**2 - b*d*x**4),x)*a**3*c*d**2*f - 
23*int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a*c + a*d*x**2 - b*c*x**2 - b* 
d*x**4),x)*a**2*b*c**2*d*f + 28*int((sqrt(c + d*x**2)*sqrt(a - b*x**2))...